Block Preconditioners Research Articles

A stochastic approach to uncertainty in the equations of MHD kinematics

The magnetohydrodynamic (MHD) kinematics model describes the electromagnetic behavior of an electrically conducting fluid when its hydrodynamic properties are assumed to be known. In particular, the MHD kinematics equations can be used to simulate the magnetic field induced by a given velocity field. While prescribing the velocity field leads to a simpler model than the fully coupled MHD system, this may introduce some epistemic uncertainty into the model. If the velocity of a physical system is not known with certainty, the magnetic field obtained from the model may not be reflective of the magnetic field seen in experiments. Additionally, uncertainty in physical parameters such as the magnetic resistivity may affect the reliability of predictions obtained from this model. By modeling the velocity and the resistivity as random variables in the MHD kinematics model, we seek to quantify the effects of uncertainty in these fields on the induced magnetic field. We develop stochastic expressions for these quantities and investigate their impact within a finite element discretization of the kinematics equations. We obtain mean and variance data through Monte Carlo simulation for several test problems. Toward this end, we develop and test an efficient block preconditioner for the linear systems arising from the discretized equations.

Algorithmically scalable block preconditioner for fully implicit shallow-water equations in CAM-SE

Performing accurate and efficient numerical simulation of global atmospheric climate models is challenging due to the disparate length and time scales over which physical processes interact. Implicit solvers enable the physical system to be integrated with a time step commensurate with processes being studied. The dominant cost of an implicit time step is the ancillary linear system solves, so we have developed a preconditioner aimed at improving the efficiency of these linear system solves. Our preconditioner is based on an approximate block factorization of the linearized shallow-water equations and has been implemented within the spectral element dynamical core within the Community Atmospheric Model (CAM-SE). In this paper, we discuss the development and scalability of the preconditioner for a suite of test cases with the implicit shallow-water solver within CAM-SE.

Splitting-based block preconditioning methods for block two-by-two matrices of real square blocks

Recently, Bai proposed rotated block preconditioners for block two-by-two matrices of real square blocks. These rotated block preconditioners have the product form of a scaled orthogonal matrix and a block two-by-two triangular matrix. Theoretical and numerical results have shown the superiority of these rotated block preconditioners (see Bai (2013) [10]). In this paper, inspired by the efficiency and the special structure of the rotated block preconditioners, we establish a new equivalent linear system to the original linear system in which an orthogonal matrix arises. We construct block Jacobi and block Gauss–Seidel splitting iteration methods based on the coefficient matrix of the new linear system. The convergence of these splitting iterations is also demonstrated. Then, by utilizing the proposed block Jacobi and block Gauss–Seidel splittings, we put forward block preconditioners which are of the product form of a scaled orthogonal matrix and a block two-by-two diagonal or a block two-by-two triangular matrix. Spectral distributions of these preconditioned matrices and numerical experiments show that the proposed splitting-based block preconditioners can be quite competitive with the rotated block preconditioners when they are used to accelerate Krylov subspace iteration methods such as GMRES for solving the block two-by-two liner systems.

On Block Preconditioners for Pde-Constrained Optimization Problems

Recently, Bai proposed a block-counter-diagonal and a block-counter-triangular preconditioning matrices to precondition the GMRES method for solving the structured system of linear equations arising from the Galerkin finite-element discretizations of the distributed control problems in (Computing 91 (2011) 379-395). He analyzed the spectral properties and derived explicit expressions of the eigenvalues and eigenvectors of the preconditioned matrices. By applying the special structures and properties of the eigenvector matrices of the preconditioned matrices, we derive upper bounds for the 2-norm condition numbers of the eigenvector matrices and give asymptotic convergence factors of the preconditioned GMRES methods with the block-counter-diagonal and the block-counter-triangular preconditioners. Experimental results show that the convergence analyses match well with the numerical results.

A new preconditioner for generalized saddle point matrices with highly singular(1,1) blocks

In this paper, based on the preconditioners presented by Cao [A note on spectrum analysis of augmentation block preconditioned generalized saddle point matrices, Journal of Computational and Applied Mathematics 238(15) (2013), pp. 109–115], we introduce and study a new augmentation block preconditioners for generalized saddle point matrices whose coefficient matrices have singular (1,1) blocks. Moreover, theoretical analysis gives the eigenvalue distribution, forms of the eigenvectors and its minimal polynomial. Finally, numerical examples show that the eigenvalue distribution with presented preconditioner has the same spectral clustering with preconditioners in the literature when choosing the optimal parameters and the preconditioner in this paper and in the literature improve the convergence of BICGSTAB and GMRES iteration efficiently when they are applied to the preconditioned BICGSTAB and GMRES to solve the Stokes equation and two-dimensional time-harmonic Maxwell equations by choosing different parameters.

Parallel Speed-Up of Preconditioned Fractional Step Navier-Stokes Solvers

Parallel performance of a fractional step Navier-Stokes solver is investigated. Parallelisation is performed using Message Passing Interface, with domain partitioning. Block preconditioning is applied to the solution of the pressure Poisson equation, which is often the bottleneck in the computation of the fractional step method. Preconditioners tested are classes of incomplete matrix decompositions and sparse approximate inverses. The computational domain is decomposed into eight parts of about equal size in terms of the number of cells, and solved on eight parallel processors. Several aspects of the parallelisation, such as domain splitting directions, speed-up and scalability of the preconditioners, are discussed.

A New Analysis of Block Preconditioners for Saddle Point Problems

We consider symmetric saddle point matrices. We analyze block preconditioners based on the knowledge of a good approximation for both the top left block and the Schur complement resulting from its elimination. We obtain bounds on the eigenvalues of the preconditioned matrix that depend only of the quality of these approximations, as measured by the related condition numbers. Our analysis applies to indefinite block diagonal preconditioners, block triangular preconditioners, inexact Uzawa preconditioners, block approximate factorization preconditioners, and a further enhancement of these preconditioners based on symmetric block Gauss--Seidel-type iterations. The analysis is unified and allows the comparison of these different approaches. In particular, it reveals that block triangular and inexact Uzawa preconditioners lead to identical eigenvalue distributions. These theoretical results are illustrated on the discrete Stokes problem. It turns out that the provided bounds allow one to localize accurately both real and nonreal eigenvalues. The relative quality of the different types of preconditioners is also as expected from the theory.

Analysis of Block Preconditioners for Models of Coupled Magma/Mantle Dynamics

This article considers the iterative solution of a finite element discretisation of the magma dynamics equations. In simplified form, the magma dynamics equations share some features of the Stokes equations. We therefore formulate, analyse and numerically test a Elman, Silvester and Wathen-type block preconditioner for magma dynamics. We prove analytically and demonstrate numerically the optimality of the preconditioner. The presented analysis highlights the dependence of the preconditioner on parameters in the magma dynamics equations that can affect convergence of iterative linear solvers. The analysis is verified through a range of two- and three-dimensional numerical examples on unstructured grids, from simple illustrative problems through to large problems on subduction zone-like geometries. The computer code to reproduce all numerical examples is freely available as supporting material.

A Block Preconditioner for an Exact Penalty Formulation for Stationary MHD

The magnetohydrodynamics (MHD) equations are used to model the flow of electrically conducting fluids in such applications as liquid metals and plasmas. This system of non-self-adjoint, nonlinear PDEs couples the Navier--Stokes equations for fluids and Maxwell's equations for electromagnetics. There has been recent interest in fully coupled solvers for the MHD system because they allow for fast steady-state solutions that do not require pseudo-time-stepping. When the fully coupled system is discretized, the strong coupling can make the resulting algebraic systems difficult to solve, requiring effective preconditioning of iterative methods for efficiency. In this work, we consider a finite element discretization of an exact penalty formulation for the stationary MHD equations posed in two-dimensional domains. This formulation has the benefit of implicitly enforcing the divergence-free condition on the magnetic field without requiring a Lagrange multiplier. We consider extending block preconditioning techniques developed for the Navier--Stokes equations to the full MHD system. We analyze operators arising in block decompositions from a continuous perspective and apply arguments based on the existence of approximate commutators to develop new preconditioners that account for the physical coupling. This results in a family of parameterized block preconditioners for both Picard and Newton linearizations. We develop an automated method for choosing the relevant parameters and demonstrate the robustness of these preconditioners for a range of the physical nondimensional parameters and with respect to mesh refinement.

The block preconditioned LSQR and GL-LSQR algorithms for the block partitioned matrices

In this paper, a new block preconditioner is proposed for the block partitioned matrices. This preconditioner is based on the block C-orthogonalization, where C is a symmetric positive definite matrix. The block preconditioned least squares (BPLS) and block preconditioned global least squares (BPGLS) algorithms are presented to solve the linear system of equations with block partioned coefficient matrix and multiple linear system of equations, respectively. The BPLS algorithm is applied to solve the complex linear system of equations and also BPLS and classical preconditioned least squares (PLS) algorithms are compared. Finally, some numerical experiments are given to show the efficiency of the new block preconditioner.

Inexact rotated block triangular preconditioners for a class of block two-by-two matrices

Based on rotated block triangular preconditioners proposed by Bai in (Sci China Math 56, 2013), we give a class of inexact rotated block triangular preconditioners for block two-by-two matrices of real square blocks $$W$$ and $$T$$ , which avoid inversion of the matrix $$\alpha W+T$$ exactly at each step of solving the linear systems. The spectral properties of the corresponding preconditioned matrices are analyzed. Numerical results show that these inexact rotated block triangular preconditioners are more effective than the exact ones when they are used to accelerate Krylov subspace iteration methods for solving block two-by-two linear systems.

Large scale micro finite element analysis of 3D bone poroelasticity

In this paper, a solver for poroelasticity problems related to osteoporotic human bones is discussed. Osteoporosis is a major health problem that compromises the integrity of bones. A good understanding of the disease requires an accurate simulation of the physics. For that purpose, a finite element solver based on Biot's consolidation equations has been developed. A mixed formulation is used to discretize the geometries taken from medical imaging. The resulting indefinite linear systems are solved by Krylov space methods supplemented by variants of Schur complement-based block preconditioners.

Accurate magnetostatic simulation of step-lap joints in transformer cores using anisotropic higher order FEM

Purpose – The purpose of this paper is to propose a solution strategy for both accurate and efficient simulation of nonlinear magnetostatic problems in thin structures using higher order finite element methods. Special interest is put in the investigation of the step-lap joints of transformer cores, with a focus on the spatial resolution of the field quantities. Design/methodology/approach – The usage of hierarchical finite elements of higher order makes it possible to adapt the local accuracy in different spatial directions in thin steel sheets. Due to explicit representation of gradients in the basis functions, a simple Schwarz-type block preconditioner with a conjugate gradient solver can efficiently solve the arising algebraic system. By adapting the block size automatically according to the aspect ratio, deterioration of convergence in case of thin elements can be prevented. The resulting Newton scheme is accelerated utilizing the hierarchical splitting in a two-level scheme, where an initial guess i...

Rotated block triangular preconditioning based on PMHSS

Based on the PMHSS preconditioning matrix, we construct a class of rotated block triangular preconditioners for block two-by-two matrices of real square blocks, and analyze the eigen-properties of the corresponding preconditioned matrices. Numerical experiments show that these rotated block triangular preconditioners can be competitive to and even more efficient than the PMHSS preconditioner when they are used to accelerate Krylov subspace iteration methods for solving block two-by-two linear systems with coefficient matrices possibly of nonsymmetric sub-blocks.

Robust partitioned block preconditioners for large-scale geotechnical applications with soil-structure interactions

SUMMARY Soil–structure interaction problems are commonly encountered in geotechnical practice and remarkably characterized with significant material stiffness contrast. When solving the soil–structure interaction problems, the employed Krylov subspace iterative method may converge slowly or even fail, indicating that the adopted preconditioning method may not suit for such problems. The inexact block diagonal preconditioners proposed recently have been shown effective for the soil–structure interaction problems; however, they haven't been exploited to full capabilities. By using the same partition strategy according to the structure elements and soil elements, the partitioned block symmetric successive over-relaxation preconditioners or partitioned block constraint preconditioners are proposed. Based on two pile-group foundation problems and a tunnel problem, the proposed preconditioners are evaluated and compared with the available preconditioners for the consolidation analysis and the drained analysis, respectively. In spite of one additional solve associated with the structure block and multiplications with off-diagonal blocks in the preconditioning step, numerical results reveal that the proposed preconditioners obviously possess better performance than the recently developed inexact block preconditioners. Copyright © 2013 John Wiley & Sons, Ltd.

Block-Symmetric and Block-Lower-Triangular Preconditioners for PDE-Constrained Optimization Problems

Optimization problems with partial differential equations as constraints arise widely in many areas of science and engineering, in particular in problems of the design. The solution of such class of PDE-constrained optimization problems is usually a major computational task. Because of the complexion for directly seeking the solution of PDE-constrained optimization problem, we transform it into a system of linear equations of the saddle-point form by using the Galerkin finite-element discretization. For the discretized linear system, in this paper we construct a block-symmetric and a block-lower-triangular preconditioner, for solving the PDE-constrained optimization problem. Both preconditioners exploit the structure of the coefficient matrix. The explicit expressions for the eigenvalues and eigenvectors of the corresponding preconditioned matrices are derived. Numerical implementations show that these block preconditioners can lead to satisfactory experimental results for the preconditioned GMRES methods when the regularization parameter is suitably small.

GPU-accelerated iterative solutions for finite element analysis of soil–structure interaction problems

Soil–structure interaction problems are commonly encountered in engineering practice, and the resulting linear systems of equations are difficult to solve due to the significant material stiffness contrast. In this study, a novel partitioned block preconditioner in conjunction with the Krylov subspace iterative method symmetric quasiminimal residual is proposed to solve such linear equations. The performance of these investigated preconditioners is evaluated and compared on both the CPU architecture and the hybrid CPU–graphics processing units (GPU) computing environment. On the hybrid CPU–GPU computing platform, the capability of GPU in parallel implementation and high-intensity floating point operations is exploited to accelerate the iterative solutions, and particular attention is paid to the matrix–vector multiplications involved in the iterative process. Based on a pile-group foundation example and a tunneling example, numerical results show that the partitioned block preconditioners investigated are very efficient for the soil–structure interaction problems. However, their comparative performances may apparently depend on the computer architecture. When the CPU computer architecture is used, the novel partitioned block symmetric successive over-relaxation preconditioner appears to be the most efficient, but when the hybrid CPU–GPU computer architecture is adopted, it is shown that the inexact block diagonal preconditioners embedded with simple diagonal approximation to the soil block outperform the others.

An efficient iterative scheme for the highly constrained augmented Stokes problem for the numerical simulation of flows in porous media

In this work, we present a new efficient iterative solution technique for large sparse matrix systems that are necessary in the mixed finite-element formulation for flow simulations of porous media with complex 3D architectures in a representative volume element. Augmented Stokes flow problems with the periodic boundary condition and the immersed solid body as constraints have been investigated, which form a class of highly constrained saddle point problems mathematically. By solving the generalized eigenvalue problem based on block reduction of the discrete systems, we investigate structures of the solution space and its subspaces and propose the exact form of the block preconditioner. The exact Schur complement using the fundamental solution has been proposed to implement the block-preconditioning problem with constraints. Additionally, the algebraic multigrid method and the diagonally scaled conjugate gradient method are applied to the preconditioning sub-block system and a Krylov subspace method (MINRES) is employed as an outer solver. We report the performance of the present solver through example problems in 2D and 3D, in comparison with the approximate Schur complement method. We show that the number of iterations to reach the convergence is independent of the problem size, which implies that the performance of the present iterative solver is close to O(N).

Preconditioning for modal discontinuous Galerkin methods for unsteady 3D Navier–Stokes equations

We compare different block preconditioners in the context of parallel time adaptive higher order implicit time integration using Jacobian-free Newton–Krylov (JFNK) solvers for discontinuous Galerkin (DG) discretizations of the three dimensional time dependent Navier–Stokes equations. A special emphasis of this work is the performance for a relative high number of processors, i.e. with a low number of elements on the processor. For high order DG discretizations, a particular problem that needs to be addressed is the size of the blocks in the Jacobian. Thus, we propose a new class of preconditioners that exploits the hierarchy of modal basis functions and introduces a flexible order of the off-diagonal Jacobian blocks. While the standard preconditioners ‘block Jacobi’ (no off-blocks) and full symmetric Gauss–Seidel (full off-blocks) are included as special cases, the reduction of the off-block order results in the new scheme ROBO-SGS. This allows us to investigate the impact of the preconditioner’s sparsity pattern with respect to the computational performance. Since the number of iterations is not well suited to judge the efficiency of a preconditioner, we additionally consider CPU time for the comparisons. We found that both block Jacobi and ROBO-SGS have good overall performance and good strong parallel scaling behavior.

Block Preconditioners for Coupled Physics Problems

Finite element discretizations of multiphysics problems frequently give rise to block-structured linear algebra problems that require effective preconditioners. We build two classes of preconditioners in the spirit of well-known block factorizations [M. F. Murphy, G. H. Golub, and A. J. Wathen, SIAM J. Sci. Comput., 21 (2000), pp. 1969--1972; I. C. F. Ipsen, SIAM J. Sci. Comput., 23 (2001), pp. 1050--1051] and apply these to the diffusive portion of the bidomain equations and the Bénard convection problem. An abstract generalized eigenvalue problem allows us to give application-specific bounds for the real parts of eigenvalues for these two problems. This analysis is accompanied by numerical calculations with several interesting features. One of our preconditioners for the bidomain equations converges in five iterations for a range of problem sizes. For Bénard convection, we observe mesh-independent convergence with reasonable robustness with respect to physical parameters, and offer some preliminary parallel scaling results on a multicore processor via message passing interface (MPI).